Yesterday we established that when the general form of a conic section has an xy-term, that the conic has been rotated and is not parallel to the x or y axis. We looked at one example (y = 1/x) and decided that the hyperbola had been rotated 45 degrees. We found the new equation yesterday by using trigonometry and the symmetry of the graph. Today will not be as easy since we can’t sketch a graph of what we are starting with.
I give the students this notes sheet and ask them what they think this graph will look like. From our lesson yesterday, most note that it will be rotated. If we want to graph, I ask them what we need to know and they realize we need to know the angle of rotation. At this point I introduce the three formulas that we will need to know in order to eliminate the xy-term and to make an equation for the x’y’ coordinate plane.
I have students find out the angle of rotation and the x and y conversions with their table group; it usually only takes a few minutes. It is a nice review of the unit circle values and we will discuss as a class to see if there are any issues.
When it is time to do the conversion, I split up the four terms of the original equation with the class. One portion will expand and simplify 7x2, while another group with simplify -6sqrt(3)xy, and another portion will simplify 13y2. I tell the class that I will simplify the -16; it’s a tough job but someone has to do it. In the video below I discuss the teaching moves to put everything together. Here is an image of our final steps.
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I tell students that this process will always allow us to graph a rotated conic in the x’y’ plane. It’s not a simple task but it will always work. My students can easily understand the process of this, but it is really easy to forget a negative sine or multiply two fractions incorrectly. I encourage them to take their time and write out their steps.
I ask them about x2 + 3xy + y2 – 6 = 0 and what type of conic this will be. Many students think it is a circle since A and B are the same value. When we graph it on Desmos they can see that it is a hyperbola. I caution students that the patterns they noticed the other day will not always hold true when there is an xy-term.
Finally, I give my students an assignment from their textbook as practice for this topic.